So, I work a lot with Magneto-Rheological (MR) fluids - which are essentially electrolyte solutions with suspended, paramagnetic colloids/particles. Once a magnetic field is applied to such a fluid, the particles tend to aggregate and line up with the external magnetic field. Here's a simulation/depiction of the structures formed from that process - where the top and bottom boundaries are electrodes:
Now, once these chains are formed, I want to apply a voltage to the electrodes and store charge by forming electric double layers around each particle. Thus, I'm looking to assess the conductivity of formed chains, and I can do this using the resistance of each sphere. Unfortunately, I have not been able to find any papers discussing an analytical expression for the resistance between two contacting spheres in water. So, I decided to implement the problem into FEA. Here's the problem: what shoould the appropriate boundary conditions be? Obviously, we're going to have a potential drop across a given set of particles, but where do we define that voltage? After all, there's going to be area in the water, and also area inside the particles (see the COMSOL domain below).
Since the voltage is unknown, I was thinking of defining the top and bottom boundaries as a function of x each and offsetting those functions by a set value. So, the bottom boundary would have V = V(x,0), and the top boundary would have V=V(x,D) + V_0, where V_0 is the difference and D is the diameter. However, I can't figure out how to implement this into COMSOL (pretty sure that you can't). So, do any of you have any idea how to implement it or have any other suggestions? Here are some simulations for the potential when V(x,0) = V_0 and V(x,D) = 0 (obviously, not the right BC).
Here's a simulation in which I set the portion inside the particles to V(x,0)=V_0 and V(x,D) = 0; then, the top/bottom boundaries for the water are periodic boundaries (closest I could get to implementing the BC described above.
THis seems more reasonable, but it's not exactly what I want. To calculate the resistance, I'm then taking I = integral of J dot n over the particle's surface area and then using dV/I = R. Any ideas / comments on how to implement the BC or a different approach to take would be GREATLY appreciated.
Note: I suppose that someone would suggest modelling the entire chain, but this is not what I want. Since I can get almost infinite shapes/chains, I'm trying to put together a method for calculating the overall resistance of a given chain on-the-fly/automatically. This would need the resistance for this case (and eventually resistance for 4 particles in one given simulation domain).